1 THE METHOD OF SUCCESSIVE APPROXIMATIONS 269

11. (a) Let F(xi9...,xp9y) be an entire function in Kp+1, such that in the power
series equal to F(;ti,..., xp, y), all monomials have a total degree ^2. Let cp be a
linear mapping of Kp into itself such that \\<p(x) \\ ^ ||jc|| for any x = (xi9...9xp)eKp;
finally, let L(x) be an arbitrary linear form on Kp. Show that there is a unique solution
/of the equation

/<*) - A/(jc/A) = F(x,f(<p(x))) (|A| > 1)
which is defined in a neighborhood of 0 and such that Km (/(*)—L(x))/\\x\\ = 0.

jc-*0

Furthermore, that solution is an entire function in Kp. (Apply Problem 10 in a neigh-
borhood of 0; reduce the problem to the case K = C, and apply (9.4.2) and (9.12.1)
to prove that / is an entire function.)

(b) Show that there is no solution of the equation f(x) — Xf(x/X) = x (A > 1) defined
in a neighborhood of 0 in R and such that f(x)/x is bounded in a neighborhood of 0.

12. Let I = [0, a]9 H= {—b, b}9 and let/be a real valued continuous function in I x H;
put M = sup |/(jc, y)\9 and let J = [0, inf(a, b/M)].

(x.y)elxH

(a) For any x e J, let E(x) be the set of values of y e H such that y = xf(x9 y).
Show that E(x) is a nonempty closed set; if g^(x) = inf(E(;t)), g2(x) = sup(E(x)),
show that gi(Q) = g2(0) = 0, and that lim 9i(x)jx= lim g2(x)/x=f(Q9G).

x-»0,x>0 jc-*0,w>0

If 9\ = 02 = 9 in J, g is continuous (cf. Section 3.20, Problem 5).

(b) Suppose a = b = 1; let E be the union of the family of the segments Sn : x = 1/2",
1/4-+1 ^ y ^ 1/4" (n ^ o), of the segments S'n: y = 1/4,1/2" ^ x ^ 1/2"-1 (n > 1) and
of the point (0,0). Define f(x,y) as follows: /(O,y) = 0; for 1/2"<x^ 1/2"'1 and
y ^ 1/4", take/(i, y) = ((y/x) + ^/((jc, ^), E))+; for 1/2" < * ^ 1/2"-1 and 1/4* ^ ^ < ;c2,
take /(jc,^) = (y/x)-tif((jc,^),E) and finally, for 1/2" < x ^ 1/2"-* and y ^ x2t
take/(x, y) — x — d((x, x2), E) (n ^ 1). Show that/is continuous, but that there is no
function g> continuous in a neighborhood of 0 in I and such that g(x) = xf(x, g(x)) in
that neighborhood.

(c) Let u0 be a continuous mapping of J into H, and define by induction iin(x) =
xf(x, wn»i(jc)) for n*z 1; the functions un are continuous mappings of J into H.
With the notations of (a), suppose that in an interval [0, c] c J, lim (un+i(x) — un(x)) =

n-*oo

0 for every xt and 0i(x) = g2(x)', show that \imun(x) = gl(x) for Q^x^c.

«-»•<»

Apply that criterion to the two following cases: (1) there exists A:>0 such that
|/(x,*i)-/(*,r2)| ^k\Zi — z2\foTxel9zl9z2 in H (compare to (10.1.1)); (2) for
0 < x ^ y ^ <3 and Zi, z2 in H, |/(x, zt) -/(x, z2)| < |^ — z2\/x.

(d) When/is defined as in (b), the sequence (un(x)) is convergent for every x e I, to a
limit which is not continuous.

(e) Take a**b~l,f(x9y)**y/x for 0 < x^ 1, \y\ ^ x2,f(x,y) = x for
0 ^ x ^ \,y^ x2,f(x9 y) « —jc for 0 ^ x ^ 1, ^ < — x2. Any continuous
function g in I such that \g(x)\ ^ x2 is a solution of #(*) = xf(x, g(x)) although
\f(x, zi) —f(x9 z2)\ < \zi — z2\/x for 0 < x < 1, zi, z2 in H; for any choice of u0, the
sequence (*/„) converges uniformly to such a solution.

(f) Define/as in (e), and let fi(x, y) = —/(#, y). The function 0 is the only solution
of g(x) — xfi(x,g(x))9 but there are continuous functions u0 for which the sequence
(un(x)) is not convergent for any x ^ 0, although \fi(x, z^ —fi(x9 z2)\ ^ zt — z2|/^ for
0<x^ I9zl9z2 inH.

13. Generalize the results of Problems 12(a) and 12(c) when H is replaced by a disk
of center (0, 0) in R2 (use the result of Problem 3 of Section 10.2).isk of center 0 and radius />„,s of C are simply connected but that their frontier